The Mathematics of Music | The Music of Mathematics "There is geometry in the humming of the strings. There is music in the motion of the spheres." --Pythagorus (in William Schaaf's Bibliography of Recreational Mathematics). The connection between mathematics and music has been appreciated for a long time. There is an intrinsic connection in the tempering of the scale and the physics of producing a tone, and in rhythmic relations. There is a more qualitative connection, too: math and music share some common themes of pattern, structure, and symmetry. The focus of this module is on the pattern and symmetry themes. Assembling the Trifold The right and left sides of the trifold are straightforward to put together. The middle section needs a bit of cutting and pasting. It's formatted so that you can print the pages out on 8 1/2 by 11 paper and have the text extend all the way across the middle section of the trifold. The Crab Canon music is printed on two lines: it needs to be cut apart and taped together so that it extends across one line. I think it's easier to figure out the symmetry if it is displayed that way. The Activities There are two main activities associated with this module, the Musical Dice Game and a counterpoint activity. Both activities assume a familiarity with reading music. This is an educational priority in the Ames elementary schools: children begin to learn to read music in first grade and most are reasonably proficient by third grade. Both activities require a xylophone or other simple instrument which is familiar to the kids. A keyboard would work also. The music for the musical dice game is provided in Printable Materials, as is a short musical phrase for the counterpoint activity. The secondary activity is to figure out what kind of symmetry Bach used in writing the Crab Canon from the Musical Offering. Each of the movements of the Musical Offering involve a mathematical transformation of the theme, but the symmetry in this one is the clearest--it is reflection symmetry. The first part is the same as the second part played backwards. Preparing the materials Materials needed: Musical Dice Game from Printable Materials, each measure glued to a separate 4x6 index card A die Enough copies of the musical phrase for each child at the event to have one A pencil Two xylophones or other kid-friendly musical instruments A printout of the Crab Canon to tape to the table in front of the display A small mirror for looking at the symmetry of the Crab Canon. Musical Dice Game: The printable materials include all the music necessary for the musical dice game. You need to supply a die. You'll need to cut out the 10 measures of music and glue them to a 4x6 index card. The Crab Canon: There's a version of the Crab Canon written out in two parts. Each part will print on an 8 1/2 x11 page. The intent is to have the music stretched across the table in a single line, first part followed by second part. It would be a good idea to have a mirror at the display, too. Doing the activities Musical dice game: Measure one has only one choice. The child rolls a die to select which choice to use for measure 2 and then lines it up next to measure 1. The child rolls a die to select which choice to use for measure 3 and then lines it up next to measure 2. The child rolls a third time to select a choice for mearure 4 and lines it up with the other 3. Then the child can play it on the xylophone. Die roll of 1 or 4 means use choice number 1, die roll of 2 or 5 means use choice 2, and die roll of 3 or 6 means use choice number 3. Counterpoint: The children can experiment with writing a second part. You need to have kids working in pairs for this activity because once they have written their accompaniment part (with any luck using some symmetry ideas) it will take two kids to play the two different parts together. Sources The Liebnitz quote on mathematics and music comes from How Math Works by Carol Vorderman (p. 61). The Crab Canon is conveniently printed in treble clef in Douglas Hofstadter's book Godel, Escher, Bach (on p. 202). The New Harvard Dictionary of Music, ed. Don Randel, contains short discussions of retrograde, inversion, and retrograde inversion (the three possible reflections in music) under the heading "Canon". There is also a discussion of aleatory music (that is, music by chance). Web Resources This website has details on the mathematical transformations which Bach used in his Musical Offering, in mathematical notation. Math and the Musical Offering by Tony Phillips This website has background information on the the Musical Offering, including the story of its origin, which is well worth reading. The Canons and Fugues of J. S. Bach by Timothy Smith You can play a musical dice game similar to Mozart's on this website: Mozart's Musikalisches Wurfelspiel by John Chuang Written Mar 19, 2002 by Janet A. Dixon Revised 1/10/06